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Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields (Probab. Theory Relat. Fields 79(4):509–542, ) model. Fix ≥1 and >0. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate (−)/, where  is the distance from the node to the root. Denote by  () the number of nodes with no descendants at time  and let  =  ln(/ln4)+(ln2)/(2). We prove that 2  ( +), ∈ℝ, converges to the Gompertz curve exp(−(ln2)  ). We also prove a central limit theorem for the martingale associated to  ().

A Markovian Growth Dynamics on Rooted Binary Trees Evolving According to the Gompertz Curve